System and method for high-frequency amplifier

ABSTRACT

A system and method for operating an amplifier system is provided. The amplifier system includes an input providing a direct coupling configured to receive a high-frequency input signal having a frequency in at least one of a radiofrequency (RF) and microwave range. The amplifier system also includes an amplifier including a dielectric material separating at least two superconducting layers forming an amplifier loop configured to receive the high-frequency input signal and deliver an amplified signal. The amplifier system includes an output providing a direct coupling configured to deliver the amplified signal.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

This invention was made with government support under FA9550-09-1-0039awarded by the USAF/AFOSR, W911 NF-10-1-0334 awarded by the ARMY/ARO,and W911NF-10-1-0324 awarded by the ARMY/ARO. The government has certainrights in the invention.

CROSS-REFERENCE TO RELATED APPLICATIONS

N/A

FIELD OF THE INVENTION

The invention relates to a system and method for a high-frequencyamplifier and, more particularly, to a system and method for a low-noisephase insensitive high-frequency amplifier.

BACKGROUND OF THE INVENTION

The rapid development of superconducting quantum electronics hasmotivated a search for quantum-limited amplifiers for the low-noisereadout of qubits and linear cavity resonators. Conventional approacheshave relied upon dc Superconducting QUantum Interference Devices (dcSQUID) that can achieve noise performance approaching the fundamentalquantum limit imposed on phase-insensitive linear amplifiers—namely, theamplifier adds at least half a quantum of noise to the signal itamplifies.

Although approaches employing SQUID are, in principle, capable ofamplifying signals at frequencies approaching the Josephson frequency(typically in the tens of GHz), it remains challenging to embed theSQUID in a traditional transmission line environment. As used herein, atransmission line or transmission line environment refers to aconductor, including a conductive trace in an integrated circuit (IC),that is specifically designed to carry alternating currents withsufficient frequencies for the signal to have properties of a wave. Suchtransmission line environments typically are a 50 Ohm (Ω) or even 75Ωenvironment. Accordingly, it is challenging to utilize a traditionalSQUID-based amplifier within a transmission line environment because itis difficult to match the inductive coupling input and the output of aSQUID-based amplifier to common transmission line environments. Forexample, the inductive coupling input of the SQUID-based amplifierpresents not only an obvious inductive component but also substantialparasitic reactance. Modeling and accounting for these characteristicsof the SQUID-based amplifier can be challenging, particularly, when theinput signal is a weak microwave tone employed as a dispersive probe ofthe quantum state of a superconducting or semiconducting qubit.

For quantum information processing applications related to the low-noisereadout of qubits and linear cavity resonators, one requiresultrasensitive amplifiers operating in the radiofrequency (RF) ormicrowave range. Using traditional amplifier designs, such asSQUID-based designs, it can be challenging to provide for efficientcoupling of an RF or microwave signal to the device. For example, theparasitic capacitance of the SQUID-based amplifier substantially impedesthe effectiveness of the amplifier architecture when utilized in the RFor microwave range. To combat the losses experienced with SQUID-basedamplifiers when used in the RF or microwave range, some have attemptedto tune the amplifier input to have a resonance matched to the expectedinput signal frequency. This strategy, though adding to the overallcomplexity of the circuit design and modeling capabilities and reducingthe flexibility of a given SQUID-based amplifier to be utilized with avaried input frequency range, often fails in the RF and microwavefrequency range due to the fact that coupling efficiency decreasessignificantly with increasing operating frequency.

Case in point, recently, it was shown that near quantum-limitedperformance can be achieved with a microstrip SQUID-based amplifier,where the input coil of the amplifier is configured as a microstripresonator with the SQUID washer acting as a groundplane. In this case,the noise temperature of the microstrip SQUID amplifier, when cooled tomillikelvin temperatures, was measured to be 47±10 mK and 48±5 mK atfrequencies of 519 MHz and 612 MHz, respectively. This performance wasmore than an order of magnitude lower than the best semiconductoramplifiers available and within a factor of 2 of the quantum limit.However, efforts to extend the operating frequencies of these amplifiersinto the gigahertz range are hampered by the fact that reduction of thelength of the input resonator is coupled to reduction of the mutualinductance between the input coil and the SQUID. Alternative approacheshave included the integration of a high-gain SQUID gradiometer into acoplanar waveguide resonator at a current antinode.

Accordingly, when used in higher-frequency applications, SQUID-basedamplifiers tend to have substantially limiting characteristics thatimpede practical implementations. Specifically, SQUID-based amplifierdevices have relatively low gain-bandwidth product in the microwavefrequency range.

Thus, it would be desirable to have a system and method for processingand amplifying readout signals having characteristics commonlyassociated with quantum computing, particularly, when the signals reachfrequencies in the RF and microwave range.

SUMMARY

The present invention overcomes the aforementioned drawbacks byproviding a system and method for a low-noise, phase-insensitive linearamplifier capable of accommodating readout signals from quantumcomputing applications, even when such signals reach frequencies in theRF and microwave range. The amplifier can significantly improve thesignal-to-noise ratio in delicate measurements of microwave signals orother high-frequency signals associated with, for example quantumcomputing or quantum communications. The amplifier incorporates alow-inductance device geometry that is relatively compact,straightforward to model at microwave frequencies, and readilyintegrated into a low-impedance RF or microwave transmission lineenvironments. Moreover, the device's input and output can both bematched to a transmission-line impedances, such as 50 or 75Ω or thelike, facilitating operation in, for example, a broadband system.

In accordance with one aspect of the invention, an amplifier system isprovided to amplify high-frequency signals. The amplifier systemincludes an input providing a direct coupling configured to receive ahigh-frequency input signal having a frequency in at least one of aradiofrequency (RF) and microwave range. The amplifier system alsoincludes an amplifier including a dielectric material separating atleast two superconducting layers forming an amplifier loop configured toreceive the high-frequency input signal and deliver an amplified signal.The amplifier system further includes an output providing a directcoupling configured to deliver the amplified signal.

In accordance with another aspect of the invention, a quantuminformation processing network is provided that is configured to receiveand relay high-frequency signals. The quantum information processingnetwork includes a signal source configured to act as at least one of aquantum computing source, a source of qubits, and a linear cavityresonator and transmits or reflects a high-frequency signal having afrequency in at least one of a radiofrequency (RF) and microwave range.The quantum information processing network also includes a transmissionline communication system configured to have an approximately 50 ohmenvironment and configured to transmit and receive the high-frequencysignal. The quantum information processing network further includes anamplifier coupled directly to the transmission line communication systemthrough an input configured to receive the high-frequency signal and anoutput to deliver an amplified high-frequency signal.

The foregoing and other aspects and advantages of the invention willappear from the following description. In the description, reference ismade to the accompanying drawings which form a part hereof, and in whichthere is shown by way of illustration a best mode and exemplaryembodiment of the invention. Such embodiment does not necessarilyrepresent the full scope of the invention, however, and reference ismade therefore to the claims and herein for interpreting the scope ofthe invention.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic illustration of a system for communicatinghigh-frequency signals in accordance with the present invention, such asmay be found in quantum information processing networks and quantumcomputing applications.

FIG. 2A is a circuit diagram of a traditional superconducting quantuminterference device (SQUID) amplifier.

FIG. 2B is a circuit diagram of an amplifier system in accordance withthe present invention.

FIGS. 3A and 3B are graphs illustrating current-voltage (I-V)characteristics of the symmetric dc SQUID and the symmetric SLUG-basedamplifier in accordance with the present invention, respectively.

FIGS. 4A and 4B are graphs illustrating the V-Φ characteristics of thesymmetric dc SQUID and the symmetric SLUG-based amplifier in accordancewith the present invention, respectively.

FIG. 5 is a graph illustrating V_(Φ) versus flux over a range of currentbias points for a SLUG-based amplifier in accordance with the presentinvention.

FIGS. 6 is a graph illustrating the relationship between R/

for a SLUG-based amplifier in accordance with the present invention overa range of bias points.

FIG. 7 is a graph illustrating the relationship between L/

for a SLUG-based amplifier in accordance with the present invention overa range of bias points.

FIG. 8 is a graph illustrating the output impedance R₀ of a SLUG-basedamplifier in accordance with the present invention over a range of biaspoints.

FIG. 9 is a graph illustrating power gain G_(m) versus flux for variousbias current I_(b) for the symmetric SLUG-based amplifier in accordancewith the present invention for an operating frequency of 5 GHz.

FIGS. 10A-10C are graphs illustrating the noises y_(V), y_(J), andy_(VJ), respectively, over a range of bias parameters of the symmetricSLUG-based amplifier in accordance with the present invention.

FIG. 11 is a simplified circuit diagram of the SLUG-based amplifier inaccordance with the present invention for purposes of evaluating devicenoise temperature T_(n).

FIG. 12 is a graph illustrating a plot of R_(s,opt)/R_(i) versus fluxfor various bias currents of the SLUG-based amplifier in accordance withthe present invention.

FIG. 13 is a graph illustrating optimal noise temperature T_(n,opt)versus flux for various bias currents I_(b) for a SLUG-based amplifierin accordance with the present invention at an operating frequencyω/2π=5 GHz.

FIGS. 14A-14C are graphs illustrating plots of the quantum noises forS_(V), S_(J), and S_(VJ), respectively, versus flux for a number of biascurrents I_(b) used with a SLUG-based amplifier in accordance with thepresent invention.

FIG. 15 is a graph illustrating R_(s,opt)/R_(i) in the quantum regimeversus flux for a range of bias I_(b) used with a SLUG-based amplifierin accordance with the present invention.

FIG. 16 is a graph illustrating n_(opt) versus flux for a variety ofmatched current sources at an operating frequency of 5 Ghz used with aSLUG-based amplifier in accordance with the present invention.

FIG. 17A is a schematic diagram illustrating a SLUG-based high-frequencyamplifier with transmission line matching sections at the input andoutput.

FIG. 17B is a schematic diagram illustrating a black box SLUG-basedamplifier for use in amplifier analysis.

FIGS. 18A-18C are graphs illustrating frequency-dependent gain, noisetemperature T_(n), and added noise quanta n for SLUG-based amplifiers inaccordance with the present invention when operated with differentsingle-section transmission line input couplers with characteristicimpedance in the range from 1-3Ω.

FIGS. 19A-19C are graphs illustrating frequency-dependent gain and noiseperformance of SLUG-based amplifiers in accordance with the presentinvention operated with different fixed single-section input couplingnetworks for gain, noise temperature in the thermal regime, and addednoise in the quantum regime for a 5 GHz signal range, respectively.

FIGS. 20A-20C are graphs illustrating frequency-dependent gain and addednoise for SLUG-based amplifiers in accordance with the present inventionoperated with different three-section matching networks.

FIG. 21A is a graph illustrating normalized gain of the SLUG-basedamplifier in accordance with the present invention versus signal powercoupled to the device input over a range of bias parameters.

FIG. 21B is a graph illustrating dynamic range of the SLUG-basedamplifier in accordance with the present invention versus flux forvarious current bias points.

FIG. 22A is a circuit diagram of a circuit model of the SLUG amplifiermodified for circuit analysis with finite input circuit admittance.

FIG. 22B is a circuit diagram of a circuit model designed to suppressthe resonances of the input by inserting a filter inductor L_(f) at theinput of the SLUG-based amplifier in accordance with the presentinvention.

FIGS. 23A are 23B are graphs illustrating the I-V and V-Φ curves of a 10pH, β_(L)=1 SLUG-based amplifier with a 10 GHz quarter-wave inputtransformer, respectively, with a transmission line input circuit withcharacteristic impedance Z₀=2Ω and bare quarter-wave resonance at 10GHz.

FIGS. 23C and 23D are graphs illustrating the I-V and V-Φ curves of a 10pH, β_(L)=1 SLUG-based amplifier with a 10 GHz quarter-wave inputtransformer, respectively, with a transmission line input circuit withcharacteristic impedance Z₀=2Φ and bare quarter-wave resonance at 10 GHzincorporating a 60 pH filter inductor.

FIG. 24A is a graph illustrating the frequency-dependent gain for theSLUG-based amplifier in accordance with the present invention.

FIG. 24B is a graph illustrating the added noise of a 5 GHz SLUG-basedamplifier calculated with the full circuit model and with the “blackbox” scattering parameters of the isolated SLUG for the SLUG circuit.

DETAILED DESCRIPTION

Referring to FIG. 1, a schematic, block diagram is provided illustratingan exemplary device architecture incorporating the present amplifiersystem. System 10 includes signal source 4. Signal source 4 may includea high-frequency signal source in the range of RF or microwave (in oneexample, a signal having a frequency between approximately 1-10 GHz).The high-frequency signal source 4 may be a quantum computing device oranother source of high-frequency communication signals, such as thosefound in satellite or space-based applications.

Signal source 4 is connected to input transmission line 6. In oneimplementation, transmission line 6 is an approximately 50 or 75Ωtransmission line. Transmission line 6 is directly connected toamplifier 8. The transmission line 6 being “directly” connected to theamplifier 8, as will be described, refers to the transmission line 6being connected to an integrated input matching network of theamplifier; this input network in turn has a direct electrical connectionto the gain element and not an inductive or other coupling. That is, aninput 9 of the amplifier 8 provides a direct coupling to thetransmission line 6 to receive the output signal of the signal source 2.As described in detail below, amplifier 8 is a low-noise,phase-insensitive linear amplifier configured to amplify a signalreceived from signal source 4 through transmission line 6.

The amplifier 8 may also be directly connected to transmission line 10that comprises a similar transmission line to that of theabove-described transmission line 6. Thus, an output 11 of the amplifier8 may be directly connected to the transmission line 10, as describedabove. For example, in one implementation, transmission line 10 is anapproximately 50 or 75Ω transmission line. Thus, as illustrated,amplifier 8 is specifically designed to be coupled with a transmissionline environment, such as illustrated by transmission line 6 andtransmission line 10. The output of amplifier 8 can then be communicatedthrough transmission line 10 to other devices and/or circuitry incommunication with system 2.

Analysis of Traditional Squid-Based Amplifier Designs

Referring now to FIG. 2A, a circuit diagram is provided illustrating asymmetric dc SQUID device 10. As shown in FIG. 2A, the gain element ofthe SQUID device consists of two overdamped Josephson junctions 12A and12B embedded in a superconducting loop 14 with inductance L. Thejunctions (with gauge invariant phases δ_(1,2)) have equal criticalcurrents I₀, self-capacitances C, and shunt resistances R. Thesuperconducting loop is formed from two equal branches, each withinductance L/2. In this model, the mutual inductance between thebranches is not considered. A dc bias current I_(b) and bias flux Φ_(b)establish a quasistatic operating point, and the signal from signalsource 4 is injected into an input coil that is coupled to the SQUIDloop 14 with mutual inductance M.

The currents through the junctions are given by equation (1):

$\begin{matrix}{{I_{1} = {{I_{0}\sin\;\delta_{1}} + \frac{\left( {V_{1} - V_{n,1}} \right)}{R} + {C\frac{\mathbb{d}V_{1}}{\mathbb{d}t}}}}{{I_{2} = {{I_{0}\sin\;\delta_{2}} + \frac{\left( {V_{1} - V_{n,2}} \right)}{R} + {C\frac{{\mathbb{d}V}\; 2_{1}}{\mathbb{d}t}}}},}} & (1)\end{matrix}$

In equation (1), V_(n,1) and V_(n,2) are noise voltages associated withthe resistive shunts, and where the voltages V_(1,2) are related to thejunction phases by the ac Josephson relation shown in equation (2):

$\begin{matrix}{{V_{1} = {\frac{\Phi_{0}}{2\pi}\frac{\mathbb{d}\delta_{1}}{\mathbb{d}t}}}{V_{2} = {\frac{\Phi_{0}}{2\pi}\frac{\mathbb{d}\delta_{2}}{\mathbb{d}t}}}} & (2)\end{matrix}$

In equation (2), Φ₀=h/2e is the magnetic flux quantum. The SQUIDamplifier 10 supports a circulating current J given by equation (3):

$\begin{matrix}{J = {\frac{I_{1} - I_{2}}{2}.}} & (3)\end{matrix}$

The voltage across the device is given by equation (4):

$\begin{matrix}\begin{matrix}{V = {V_{1} + {\frac{L}{2}\frac{\mathbb{d}I_{1}}{\mathbb{d}t}}}} \\{= {V_{2} + {\frac{L}{2}{\frac{\mathbb{d}I_{2}}{\mathbb{d}t}.}}}}\end{matrix} & (4)\end{matrix}$

The circulating current and the junction phases are related to the totalflux in the loop Φ_(T) as shown in equation (5).

$\begin{matrix}\begin{matrix}{\Phi_{T} = {\Phi_{b} + {LJ}}} \\{= {\frac{\Phi_{0}}{2\pi}{\left( {\delta_{2} - \delta_{1}} \right).}}}\end{matrix} & (5)\end{matrix}$

In the analysis of the present SQUID device model 10, the dimensionlessvariables (i, v, Φ, and θ) are defined as follows: i≡I/I_(o),v≡V/I_(o)R, Φ≡Φ/Φ₀, and θ≡t/[Φ₀/2πI₀R)]. In addition, the dimensionlessreduced inductance C_(L)=2I₀L/Φ₀ and the damping parameterβ_(C)=(2π/Φ₀)I₀R²C. The equations of motion for the junction phases areas shown in equation (6).

$\begin{matrix}{{{\beta_{C}{\overset{¨}{\delta}}_{1}} = {\frac{i_{b}}{2} + \frac{\delta_{2} - \delta_{1} - {2{\pi\phi}_{b}}}{{\pi\beta}_{L}} - {\sin\;\delta_{1}} - {\overset{.}{\delta}}_{1} + v_{n,1}}}{{\beta_{C}{\overset{¨}{\delta}}_{2}} = {\frac{i_{b}}{2} + \frac{\delta_{2} - \delta_{1} - {2{\pi\phi}_{b}}}{{\pi\beta}_{L}} - {\sin\;\delta_{2}} - {\overset{.}{\delta}}_{2} + {v_{n,2}.}}}} & (6)\end{matrix}$

The quasistatic output voltage and circulating current are given byequations (7) and (8), respectively.

$\begin{matrix}{v_{out} = {\frac{1}{2}\left( {{\overset{.}{\delta}}_{1} + {\overset{.}{\delta}}_{2}} \right)}} & (7) \\{j = {\frac{1}{{\pi\beta}_{L}}{\left( {\delta_{2} - \delta_{1} - {2{\pi\phi}_{b}}} \right).}}} & (8)\end{matrix}$

EXAMPLE AMPLIFIER GEOMETRY AND ANALYSIS

Referring now to FIG. 2B, a circuit diagram is provided illustrating anamplifier system 16 that, in accordance with the present disclosure,includes a low-noise, phase-insensitive linear amplifier 8 capable ofaccommodating readout signals from quantum computing applications, evenwhen such signals reach frequencies in the RF and microwave range. Asshown in FIG. 2B, the amplifier device 8 includes a loop 18 formed fromtwo superconducting traces 20, 22 separated by a thin dielectric layer24. At an input 26, the loop 18 is connected to a transmission lineenvironment 6. A signal source 4, which as described above, may be aquantum computing device or another source of high-frequencycommunication signals, such as those found in satellites or the like,injects a signal having an RF, microwave, or similar high-frequencysignal directly into the transmission line environment and, thereby,directly to the amplifier 8.

It is contemplated that the amplifier 8 may have a general architecturereferred to herein as a Superconducting Low-inductance UndulatoryGalvanometer (SLUG). It is noted that the original work relating tosuperconducting low-inductance undulatory galvanometers (SLUGs) wasfocused on the creation of sensors by arranging a bead of PbSn solderfrozen on a Nb wire to thereby measure currents passed along the Nb wireby way of currents measured between the two superconductors formed bythe frozen solder. The current between the superconductors is periodicand changes therein can be readily measured or, the low inductance atthe input to the sensor can be exploited to create a voltage sensorusing substantially small resistances. As used herein, a SLUG-basedamplifier refers to an architecture including the dielectric materials24 (e.g., one or more thin film dielectric materials may be used)separating the superconducting layers 20, 22. By way of thisarchitecture, the SLUG-based amplifier 8 achieves an extremely smalleffective area (for example, by using a thin dielectric layer to formthe loop) and input coupling by direct current injection as opposed toflux coupling. As described above, the input to the SLUG-based amplifieris by direct current injection, as opposed to, for example, an inductivecoupling found in a SQUID and other amplifier designs. Inimplementations where the amplifier device 8 is integrated into amicrostrip transmission line, the amplifier device 8 can be realized inthree metallization steps (corresponding to the circuit ground-plane andthe two arms of the SLUG), with two dielectric thin films separating themetal layers. The mutual inductance between the arms of the amplifierdevice 8 is on the order of the self-inductance of the arms, and must betaken into account.

The total flux through the amplifier device 8 is shown in equation (9).

$\begin{matrix}\begin{matrix}{\Phi_{T} = {{L\left( {I_{1} + I_{\Phi}} \right)} + \Phi_{b}}} \\{= {\frac{\Phi_{0}}{2\pi}{\left( {\delta_{2} = \delta_{1}} \right).}}}\end{matrix} & (9)\end{matrix}$

For the amplifier device 8, the dimensionless equations of motion forδ_(1,2) are shown in equation (10).

$\begin{matrix}{{{\beta_{C}{\overset{¨}{\delta}}_{1}} = {\frac{\delta_{2} - \delta_{1} - {2{\pi\phi}_{b}}}{{\pi\beta}_{L}} - i_{\phi} - {\sin\;\delta_{1}} - {\overset{.}{\delta}}_{1} + v_{N,1}}}{{\beta_{C}{\overset{¨}{\delta}}_{2}} = {\frac{\delta_{2} - \delta_{1} - {2{\pi\phi}_{b}}}{{\pi\beta}_{L}} + i_{b} + i_{\phi} - {\sin\;\delta_{2}} - {\overset{.}{\delta}}_{2} + v_{N,2}}}} & (10)\end{matrix}$

The output voltage and circulating current are given by equations (11)and (10), respectively.

$\begin{matrix}{v_{out} = {\overset{.}{\delta}}_{2}} & (11) \\{j = {{\frac{1}{{\pi\beta}_{L}}\left( {\delta_{2} - \delta_{1} - {2{\pi\phi}_{b}}} \right)} - {i_{\phi}/2.}}} & (12)\end{matrix}$

To operate the SQUID-based amplifier described above with respect toFIG. 2B or the amplifier device 8 of FIG. 2B as an amplifier, I_(b) andΦ_(b) are chosen to establish a quasistatic operating point where thetransfer function VΦ≡δ_(V)/δ_(Φ)is large.

In both cases, the device acts as a transimpedance element. That is, thesignal from the signal source 4 is coupled to the device as a current,and the output signal is coupled from the device as a voltage.

DC Characteristic Analysis

In example implementations, equations (6) and (10) are numericallyintegrated using a 4th order Runge-Kutta solver for N˜2¹⁸ time steps Δθover a range of bias points.

FIGS. 3A and 3B are graphs illustrating the current-voltage (I-V)characteristics of the symmetric dc SQUID and the symmetric SLUG withβ_(L)=1 and β_(C)=0.8, respectively. In FIG. 3A, three lines are shownrepresenting Φ_(b)=0Φ₀ (line 30), Φ_(b)=0.25Φ₀(line 32), andΦ_(b)=0.5Φ₀(line 34). In FIG. 3B, three lines are shown representingΦ_(b)=0Φ₀ (line 36), Φ_(b)=0.25Φ₀ (line 38), and Φ_(b)=0.5Φ₀ (line 39).

FIGS. 4A and 4B are graphs illustrating the V-Φ characteristics of thesymmetric dc SQUID and the symmetric SLUG with β_(L)=1 and β_(C)=0.8,respectively. In FIG. 4A, three lines are shown representing I_(b)=1.8I₀(line 40), I_(b)=1.9 I₀ (line 42), and I_(b)=2.0 I₀ (line 44). In FIG.4B, three lines are shown representing I_(b)=1.8 I₀ (line 46), I_(b)=1.9I₀ (line 48), and I_(b)=2.0 I₀ (line 49).

For bias near 1.9 I₀, the peak-to-peak voltage modulation is somewhatabove 0.5 I₀R. The dc characteristics of the SLUG amplifier closelymatch those of the symmetric dc SQUID, apart from a shift in flux biaspoint that arises from the asymmetric division of the bias current tothe amplifier of the present invention. Similarly, the scatteringparameters and noise properties of the amplifier of the presentinvention and the SQUID are closely matched, apart from this bias shift.

In the following analysis of amplifier devices created in accordancewith the present invention, the following set of SLUG parameters areused: β_(L)=1, β_(C)=0.8, L=10 pH, and C=50 fF, corresponding to ajunction with critical current 100 microamps (μA) and area around 1micro square meter (μm²). These device parameters are examples ofappropriate parameters for working SLUG devices for at least thefollowing reasons. First, in manufacturing such a device, inductances onthe order of 10 pH are realized in a reliable, controlled way using theSLUG geometry, and the resulting device is generally less susceptible tostray reactances and is straightforward to model at microwavefrequencies. The required critical current density is 10 kA/cm2, whichcan be provided using standard Nb—AlO_(x)—Nb technology. While Jouleheating in the shunt resistors may be appreciable, the addition oflarge-volume normal-metal cooling fins can allow equilibration of theshunt resistors at temperatures below 100 mK. Lower device inductancesmay require uncomfortably high junction critical currents to achievecomparable device performance, and fabrication yield and Joule heatingof the shunts could become less desirable. Alternatively, asignificantly larger SLUG inductance can be used at the cost ofproviding less gain and complicating the microwave engineering, owing tothe larger device dimensions.

Scattering Parameters

In order to optimize SLUG amplifier design in accordance with thepresent invention, it is useful to understand the forward transferfunction and the complex input and output impedances of the device. Toextract these from the present SLUG device model, appropriate testsignals can be applied to the device or model, followed by probing ofthe complex response at the excitation frequency, chosen to be a smallfraction of the Josephson frequency ω_(J)/2π. The forward transimpedanceV_(I)≡δV/δI is readily derived from the SLUG flux-to-voltage transferfunction VΦ:V_(I=MV) _(Φ′)  (13)

In equation (13), M=L for the case of the symmetric SLUG.

FIG. 5 is a graph illustrating SLUG V_(Φ) versus flux over a range ofcurrent bias points for β_(L)=1 and β_(C)=0.8. In FIG. 5, three linesare shown representing I_(b)=1.8 I₀ (line 50), I_(b)=1.9 I₀ (line 52),and I_(b)=2.0 I₀ (line 54).

Next, consider the input return loss. The SLUG input in accordance withthe present invention is an inductive short to ground at lowfrequencies, and the complex input impedance Z_(i) is frequencydependent. The input impedance is readily derived from the dynamicimpedance Z, defined in terms of the flux-to-current transfer functionJ_(Φ)≡δJ/δΦ, as shown in equation (14).

 - J Φ ≡ 1 = 1 + jω , ( 14 )

where

is the frequency-independent dynamic resistance, while

is the dynamic inductance. FIGS. 6 and 7 are graphs illustrating therelationship between R/

and L/

respectively, for a SLUG device having β_(L)=1 and β_(C)=0.8 over arange of bias points. In FIG. 6, three lines are shown representingI_(b)=1.8 I₀ (line 60), I_(b)=1.9 I₀ (line 62), and I_(b)=2.0 I₀ (line64). In FIG. 7, three lines are shown representing I_(b)=1.8 I₀ (line70), I_(b)=1.9 I₀ (line 72), and I_(b)=2.0 I₀ (line 74).

FIG. 8 is a graph illustrating the SLUG device output impedance R₀ overa range of bias points. In FIG. 8, three lines are shown representingI_(b)=1.8 I₀ (line 80), I_(b)=1.9 I₀ (line 82), and I_(b)=2.0 I₀ (line84).

The output impedance of the SLUG device is real and frequencyindependent, and the magnitude of R₀ is on the order of the junctionshunt resistance R.

For the following discussion, it is convenient to work in terms of thebias-dependent dimensionless impedance parameters ρ_(i,o), defined byequation (15).

$\begin{matrix}{{R_{i} = {\rho_{i}\frac{\left( {\omega\; M} \right)^{2}}{R}}}{R_{0} = {\rho_{0}R}}} & (15)\end{matrix}$

From the definition of

above, it follows that ρ_(i)=R/

Amplifier gain, bandwidth, and noise properties depend sensitively onp_(i) and p_(o).

Power gain of the SLUG device, in one implementation, is maximized whenappropriate conjugate matching networks are employed to couple thesignal to and from the device. The maximum available power gain G_(m) isprovided by equation (16).

$\begin{matrix}{G_{M} = {\frac{{V_{0}^{2}/4}\; R_{0}}{I_{i}^{2}R_{i}}.}} & (16)\end{matrix}$

In equation (16), I_(i) is the input current and V_(o) is the outputvoltage.

Using equation (15), equation (17) follows:

$\begin{matrix}{G_{M} = {\frac{1}{4\rho_{i}\rho_{0}}{\left( \frac{V_{\Phi}}{\omega} \right)^{2}.}}} & (17)\end{matrix}$

FIG. 9 is a graph illustrating power gain G_(m) versus flux for variousbias current I_(b) for the symmetric SLUG device with β_(L)=1,β_(C)=0.8, L=10 pH, and C=50 fF for an operating frequency in excess of5 GHz. In FIG. 9, three lines are shown representing I_(b)=1.8 I₀ (line90), I_(b)=1.9 I₀ (line 92), and I_(b)=2.0 I₀ (line 94). Over a broadrange of bias parameters, gain in excess of 20 dB is achievable. It isimportant to note, however, that a conjugate match to a 50Ω source doesnot readily yield best amplifier noise performance, due to the mismatchbetween the real part of the SLUG input impedance R_(i) and the optimalnoise-matched source impedance, which can be significantly larger thanR_(i). Amplifier optimization therefore involves a trade-off betweengain and noise performance, as discussed in detail below.

The bandwidth of the SLUG amplifier will be determined by the couplingto the low-impedance input port, as the device output is reasonablywell-matched to typical transmission line impedances. In the followingdiscussion, to generate an estimate of amplifier bandwidth, a 50Ω sourceimpedance is considered. However, such discussion will illustrate thatother similar source impedances can be accommodated. It is assumed thatconjugate matching at the SLUG device input is accomplished via aquarter-wave transmission line section. For discussion purposes, thepresent disclosure neglects the imaginary part of the SLUG inputimpedance. The amplifier quality factor Q in that case is given byequation (18).

$\begin{matrix}{{Q \approx {\frac{\pi}{8}\sqrt{\frac{50\Omega}{R_{i}}}}} = {\frac{\pi}{8\omega\; M}{\sqrt{\frac{50\Omega \times R}{\rho_{i}}}.}}} & (18)\end{matrix}$

The bandwidth of an amplifier designed at an operating frequency ω/2π isthen ω/2πQ. For an operating frequency around 5 GHz, R_(i) is on theorder of 0.1Ω. Therefore Q is expected to be on the order of 10, andamplifier bandwidths on the order of hundreds of MHz. For current biasI_(b)<2I₀ and for a narrow range of fluxes corresponding to bias pointsnear the supercurrent branch, it is possible to achieve extremely highpower gain (see, for example, FIG. 9). However, the high gains achievedat these bias points are due largely to vanishing R_(i) 13 an amplifierdesigned to operate in this regime would have a rather small bandwidth.It is important to note that equation (18) presents only a roughguideline for the bandwidth rather than a fundamental limit. Inparticular, it is possible to obtain a larger bandwidth with less or nodegradation in gain by employing either a tapered transmission linematching section or a multisection input transformer with steppedtransmission line impedances.

Noise Properties in the Thermal Regime

The Johnson noise of the shunt resistors gives rise to a voltage noiseat the device output and to a circulating current noise in the deviceloop; moreover, these noises are partially correlated, since thecirculating current noise couples a flux noise to the loop, which inturn yields a voltage noise across the device. To incorporate noise inthe present model, a pseudorandom number generator can be used to createa gaussian-distributed set of voltages v_(N,1) and v_(N,2) with zeromean and variance 2Γ/Δθ, with the dimensionless noise parameterΓ=2πk_(B)T/I₀Φ₀, corresponding to the usual white power spectral densityS_(v)=4Γ for Johnson noise in the thermal limit. The simulations wereaveraged over many (˜100) realizations of the random noise voltages.

The dimensionless noise parameters y_(V), y_(J), and y_(VJ), such thatthe voltage noise spectral density at the device output is given byS_(V)=2y_(V)k_(B)TR, the circulating current noise spectral density isSJ=2y_(J)k_(B)T/R, and the cross noise spectral density isS_(VJ)=2y_(VJ)k_(B)T; here T is the electron temperature of the shuntresistors. These noises are calculated by solving the Langevin equations(10).

The noise spectrum consists of a series of peaks at the Josephsonfrequency and its harmonics; the dimensionless noises are evaluated atlow frequency f<<ω_(J)/2π where the spectrum is white. The noises y dodepend on the noise parameter Γ, due to the possibility of saturationand smearing of the device characteristics at elevated temperature.

FIGS. 10A-10C are graphs illustrating the dimensionless noises y_(V),y_(J), and y_(VJ), respectively, over a range of bias parameters of thesymmetric SLUG device for device parameters β_(L)=1, ⊕_(C)=0.8, andΓ=4×10⁻⁵; this choice corresponds to a temperature of 100 mK for ajunction critical current of 100 μA. In FIGS. 10A-10C, for each graph,three lines are shown representing I_(b)=1.8 I₀ (line 100), I_(b)=1.9 I₀(line 102), and I_(b)=2.0 I₀ (line 104). At high bias current I_(b)>>I₀,y_(V,J) approaches the expected Johnson noise limit of 1 (one) for thetwo shunt resistors in parallel.

The device noise temperature T_(n) can be evaluated from the circuitshown in FIG. 11. There it is assumed a noiseless source impedanceZ_(s)=R_(s)+jX_(s) and equate the total noise of the amplifier to thenoise contribution from a source resistance R_(s) at an effectivetemperature T_(n). All noises are referred to the device output. Thisresults in equation (19).

$\begin{matrix}{{4\; k_{B}T_{n}R_{s}\frac{V_{\Phi}^{2}M^{2}}{R_{t}^{2} + X_{t}^{2}}} = {{2\gamma_{V}k_{B}{TR}} + {\frac{{2\gamma_{J}k_{B}T}\;}{R}\frac{\omega^{2}V_{\Phi}^{2}M^{4}}{R_{t}^{2} + X_{t}^{2}}} + {4\gamma_{VJ}k_{B}T{\frac{\omega\; V_{\Phi}M^{2}X_{t}}{R_{t}^{2} + X_{t}^{2}}.}}}} & (19)\end{matrix}$

In equation (19) R_(t)=R_(s)+R_(i)(X_(t)=X_(s)+X_(i)) is the sum of thereal (imaginary) parts of the source impedance and the device inputimpedance. The noise temperature is thus given by equation (20).

$\begin{matrix}{T_{n} = {\left\lbrack {{\frac{\gamma_{V}}{2}\frac{\left( {R_{t}^{2} + X_{t}^{2}} \right)R}{V_{\Phi}^{2}M^{2}R_{S}}} + {\frac{\gamma_{J}}{2}\frac{\left( {\omega^{2}M^{2}} \right)}{{RR}_{S}}} + {\gamma_{VJ}\frac{\omega\; X_{t}}{V_{\Phi}R_{S}}}} \right\rbrack{T.}}} & (20)\end{matrix}$

The condition δT_(n)/δX_(t)=0 is used to solve for the imaginary part ofthe optimal source impedance, as shown in equation (21).

$\begin{matrix}{{X_{s,{opt}} = {{{- \frac{\gamma_{VJ}}{\gamma_{V}}}\frac{\omega\; V_{\Phi}M^{2}}{R}} - X_{i}}},} & (21)\end{matrix}$

Similarly, the condition δT_(n)/δR_(s)=0 yields the real part of theoptimal source impedance as shown in equation (22).

$\begin{matrix}{R_{s,{opt}} = {\left\lbrack {1 + {\frac{1}{\gamma_{V}^{2}\rho_{i}^{2}}\left( \frac{V_{\Phi}}{\omega} \right)^{2}\left( {{\gamma_{V}\gamma_{J}} - \gamma_{VJ}^{2}} \right)}} \right\rbrack^{1/2}{R_{i}.}}} & (22)\end{matrix}$

For bias points where V_(Φ) is highest, equation (23) shows anapproximate expression for R_(s,opt).

$\begin{matrix}{{R_{s,{opt}} \approx {\frac{1}{\gamma_{V}\rho_{i}}\frac{V_{\Phi}}{\omega}\left( {{\gamma_{V}\gamma_{J}} - \gamma_{VJ}^{2}} \right)^{\frac{1}{2}}R_{i}}} = {\frac{\omega\; V_{\Phi}M^{2}}{\gamma_{V}R}{\left( {{\gamma_{V}\gamma_{J}} - \gamma_{VJ}^{2}} \right)^{1/2}.}}} & (23)\end{matrix}$

FIG. 12 is a graph illustrating a plot of R_(s,opt)/R_(i) versus fluxfor various bias currents. In FIG. 12, three lines are shownrepresenting I_(b)=1.8 I₀ (line 110), I_(b)=1.9 I₀ (line 112), andI_(b)=2.0 I₀ (line 114).

For typical device parameters, R_(s,opt)>>R_(i). For this reason, it canbe difficult to achieve a simultaneous power match and noise match. Itis worthwhile to note, however, that the ratio R_(s,opt/)R_(i) scaleswith frequency as ω⁻¹, facilitating simultaneous attainment of high gainand good noise performance at higher operating frequencies.

When the signal is coupled to the device via a source with optimalimpedance R_(s,opt)+jX_(s,opt), the amplifier noise temperature becomes:

$\begin{matrix}{T_{n,{opt}} = {\frac{\omega}{V_{\Phi}}\left( {{\gamma_{V}\gamma_{J}} - \gamma_{VJ}^{2}} \right)^{1/2}{T.}}} & (24)\end{matrix}$

FIG. 13 is a graph illustrating optimal noise temperature T_(n,opt)versus flux for various bias currents I_(b) for a SLUG amplifier at anoperating frequency ω/2π=5 GHz. In FIG. 12, three lines are shownrepresenting I_(b)=1.8 I₀ (line 120), I_(b)=1.9 I₀ (line 122), andI_(b)=2.0 I₀ (line 124). Note that every point in these plotscorresponds to a different realization of the input matching network.

Noise Properties in the Quantum Regime

At sufficiently low temperature, the zero-point fluctuations of theresistive shunts are expected to make the dominant noise contribution.The full expression for the spectral density of voltage noise producedby the resistors is written as 2hfRcoth(hf/2k_(B)T). The added noise ofthe symmetric SLUG in the zero-temperature limit, where the voltagespectral density of the shunt resistors becomes 2hfR. A single-sidedquantum spectral density can be calculated by digitally filteringgaussian white noise. Using the quantum noise as a driving term in theLangevin equations (10), the voltage power spectral density S_(V)(f) atthe device output, the circulating current spectral density S_(J)(f),and the cross spectral density S_(VJ)(f) are evaluated.

FIGS. 14A-14C are graphs illustrating plots of the quantum noises forS_(V), S_(J), and S_(VJ), respectively, versus flux for a number of biascurrents I_(b). In FIGS. 14A-14C, for each graph, three lines are shownrepresenting I_(b)=1.8 I₀ (line 130), I_(b)=1.9 I₀ (line 132), andI_(b)=2.0 I₀ (line 134). Once again, the device noise temperature T_(n)can be evaluated from the circuit of FIG. 11. A zero-temperature sourceimpedance Zs=Rs+jXs is assumed and the total noise of the amplifier isequated to the noise contribution from a source resistance R_(S) at afinite effective temperature T_(n). The amplifier noise temperature isobtained from equation (25).

$\begin{matrix}{{2\;{hfR}_{S}{\coth\left( {{{hf}/2}\; k_{B}T_{n}} \right)}\frac{V_{\Phi}^{2}M^{2}}{R_{t}^{2} + X_{t}^{2}}} = {S_{V} + {S_{J}\frac{\omega^{2}\; V_{\Phi}^{2}M^{4}}{R_{t}^{2} + X_{t}^{2}}} + {2\; S_{VJ}\frac{\omega\; V_{\Phi}M^{2}X_{t}}{R_{t}^{2} + X_{t}^{2}}} + {2\;{hfR}_{S}{\frac{V_{\Phi}^{2}M^{2}}{R_{t}^{2} + X_{t}^{2}}.}}}} & (25)\end{matrix}$

Alternatively, it is possible to express the noise contribution of thedevice in terms of an added number of noise photons n, where n and T_(n)are related as shown in equation (26).coth(hf/2k _(B) T _(n))=2n+1   (26)

In accordance with equation (26), equation (27) is determined.

$\begin{matrix}{n = {\frac{1}{2\;{hfR}_{S}}\left\lbrack {{\frac{S_{V}}{2}\frac{R_{t}^{2} + X_{t}^{2}}{V_{\Phi}^{2}M^{2}}} + {\frac{S_{J}}{2}\omega^{2}M^{2}} + {S_{VJ}\frac{\omega}{V_{\Phi}}X_{t}}} \right\rbrack}} & (27)\end{matrix}$

The optimal source impedance Z_(s,opt)=R_(S,opt)+jX_(s,opt) is obtainedfrom the relations δ_(n)/δX_(t)=0 and δ_(n)/δR_(S)=0. The imaginary partof the optimal source impedance is given by equation (28).

$\begin{matrix}{X_{s,{opt}} = {{{- \frac{S_{VJ}}{S_{V}}}\omega\; V_{\Phi}M^{2}} - x_{i}}} & (28)\end{matrix}$

Similarly, the real part of the optimal source impedance is shown byequation (29).

$\begin{matrix}{R_{s,{opt}} = {\left\lbrack {1 + {\left( \frac{V_{\Phi}R}{\rho_{i}\omega\; S_{V}} \right)^{2}\left( {{S_{V}S_{J}} - S_{VJ}^{2}} \right)}} \right\rbrack^{1/2}{R_{i}.}}} & (29)\end{matrix}$

In the limit V_(Φ)>>ω, equation (30) follows.

$\begin{matrix}{R_{s,{opt}} \approx {\frac{\omega\; V_{\Phi}M^{2}}{S_{V}}\left( {{S_{V}S_{J}} - S_{VJ}^{2}} \right)^{1/2}}} & (30)\end{matrix}$

FIG. 15 is a graph illustrating R_(s,opt)/R_(i) in the quantum regimeversus flux for a range of bias I_(b). In FIG. 14, three lines are shownrepresenting I_(b)=1.8 I₀ (line 140), I_(b)=1.9 I₀ (line 142), andI_(b)=2.0 I₀ (line 144). FIG. 15 depicts impedance at any operatingfrequency of 5 Ghz with SLUG device parameters of β_(L)=1, β_(C)=0.8,L=10 pH, and C=50 fF.

For the optimally matched source, the added number of noise photons isgiven by equation (31).

$\begin{matrix}{n_{opt} = {\frac{1}{2\;{hV}_{\Phi}}\left( {{S_{V}S_{J}} - S_{VJ}^{2}} \right)^{1/2}}} & (31)\end{matrix}$

FIG. 16 is a graph illustrating n_(opt) versus flux for a variety ofmatched current sources at an operating frequency of 5 GHz, with SLUGdevice parameters of β_(L)=1, β_(C)=0.8, L=10 pH, and C=50 fF. In FIG.16, three lines are shown representing I_(b)=1.8 I₀ (line 150),I_(b)=1.9 I₀ (line 152), and I_(b)=2.0 I₀ (line 154). As shown by FIG.15, for an appropriately noise-matched source, the SLUG approaches anoise level that is close to the standard quantum limit n_(SQL)=½, aminimum achievable added noise for a phase-insensitive linear amplifier.

Amplifier Design Using Analysis Constraints

As shown by the above analysis, the above-described SLUG-based amplifierdesign is an attractive gain element for the realization of a low-noise,high-frequency, such as RF or microwave frequency, amplification. In thepresent system, the SLUG device can be incorporated into transmissionline environments, such as existing, external networks used to embed thedevice in a 50Ω environment. In doing so, the design is configured todeliver desirable power transfer to and from the device and tosubstantially match the impedance (e.g., 50Ω) of the source to thedesired noise impedance at the desired operating frequency.

For example, to maximize gain, a conjugate matching network may bedesigned to transform the 50Ω source to R_(i)-jX_(j). On the other hand,optimal noise performance may be achieved for an input matching networkthat transforms the 50 Ω generator to the complex optimal sourceimpedance Z_(s,opt)=R_(s,opt)+jX_(s,opt). Because R_(s,opt)>>R_(i) fortypical parameters, it is generally difficult to achieve a simultaneouspower match and noise match. However, it can be possible to find acompromise where there is reasonable gain and good noise performanceover a relatively broad bias range. FIG. 17A is a schematic diagramshowing a SLUG-based high-frequency amplifier 160 with transmission linematching sections at the input 162 and output 164.

To calculate amplifier gain and noise performance, the SLUG-basedamplifier 160 is treated as a “black box” with scattering and noiseparameters derived from the calculations described above. FIG. 17B is aschematic diagram showing the black box SLUG-based amplifier 160 for usein amplifier analysis.

As an example, FIGS. 18A-18C are graphs illustrating frequency-dependentgain, noise temperature T_(n), and added noise quanta n for SLUG-basedamplifiers operated with different single-section transmission lineinput couplers with characteristic impedance in the range from 1-3Ω. InFIGS. 18A-18C, for each graph, three lines are shown representingI_(b)=1.8 I₀ and an impedance of 1Ω (line 170), I_(b)=1.9 I₀ and animpedance of 2Ω (line 172), and I_(b)=2.0 I₀ and an impedance of 3 Ω(line 174).

Here, the full expressions (20) and (27) are used to calculate thefrequency-dependent noise contribution of the amplifier in the thermaland quantum regimes, respectively. The length of the input couplerprovides a bare quarter-wave resonance at 6.5 GHz; inductive loading bythe SLUG-based amplifier pulls the operating frequency down to thedesired value of 5 GHz. Note that the transmission line impedancesconsidered here are readily achieved with thin-film microstriptechnology: for example, a trace width of 10 μm and a dielectric withε_(r)=4 and thickness 100 nm corresponds to a characteristic impedanceof 2Ω.

FIGS. 19A-19C are graphs illustrating frequency-dependent gain and noiseperformance of SLUG amplifiers operated with different fixedsingle-section input coupling networks for gain, noise temperature inthe thermal regime, and added noise in the quantum regime for a 5 GHzSLUG amplifier, respectively. In FIGS. 19A-19C, for each graph, threelines are shown representing 0.25Φ₀ (line 180), 0.3Φ₀ (line 182), and0.35Φ₀ (line 184). In FIGS. 19A-19C, the input matching network is asingle transmission line section with bare quarter-wave resonance at 6.5GHz and characteristic impedance of 2Φ with device parameters ofβ_(L)=1, β_(C)=0.8, L=10 pH, C=50 fF, and I_(b)=1.8I₀.

Due to the non-vanishing cross spectral density S_(VJ), the minimumnoise temperature occurs at a frequency that is somewhat lower than thefrequency of maximum gain. For a Z_(0,i)=2Φ input coupler, noise within50% of the standard quantum limit can be achieved at a frequency whereamplifier gain is 15 dB, and noise within a factor of 2 of the standardquantum limit at a frequency where gain is 18 dB.

Further, it is possible to increase amplifier bandwidth significantly bycoupling the input signal to the device via a multi-section transformerwith stepped characteristic impedances. As an example, FIG. 20A-20C aregraphs illustrating the frequency-dependent gain, noise temperature inthe thermal regime, and added noise in the quantum regime, respectively,for broadband amplifiers incorporating a 10 pH SLUG element. The devicecharacteristics are β_(L)=1, β_(C)=0.8, I_(b)=1.8I₀, and Φ_(b)=0.35ΦI₀.FIGS. 20A-20C illustrate frequency-dependent gain and added noise foramplifiers operated with different three-section matching networks. Ineach of FIGS. 20A-20C two lines are shown. Line 190 illustrates athree-section input matching network with quarter-wave resonances at 5GHz and with characteristic impedances of 24.3Ω, 17.4Ω, and 3.0Ω derivedfrom numerical minimization of the SLUG quantum noise over the band from4.5 GHz to 5.5 GHz. Line 192 illustrates a matching network consistingof three sections with characteristic impedance 29.8Ω, 7.1Ω, and 1.1Ωfollowed by a series capacitance of 38 pF to tune out the imaginary partof the input impedance to the amplifier at a frequency of 5 GHz. Withthese exemplary operating parameters, the length of each transmissionline section may be chosen to provide a bare quarter-wave resonance at 5GHz, and the characteristic impedances determined by numericalminimization of the quantum noise contribution of the SLUG amplifier inthe frequency range from 4.5 to 5.5 GHz.

Dynamic Range

The strong nonlinearity of the SLUG amplifier device leads to gaincompression and harmonic generation when the amplifier device is drivenwith a large-amplitude signal. It is important to verify that thedynamic range of the amplifier device will be sufficient for the desiredapplication. FIG. 21A is a graph illustrating normalized gain of theabove-described amplifier device in accordance with the presentinvention versus signal power coupled to the device input over a rangeof bias parameters for β_(L)=1, β_(C)=0.8, L=10 pH and C=50 fF. FIG. 21Ashows 5 (five) lines corresponding to 0.225Φ₀ (line 200), 0.25Φ₀ (line201), 0.275Φ₀ (line 202), 0.3Φ₀ (line 203), and 0.325Φ₀ (line 204). Theplots of FIG. 21A were generated by solving the SLUG equations of motion(i.e., equation (10)) with a sinusoidal driving term of varyingamplitude. Depending on bias point, the 1 dB compression point occurssomewhere in the range from −110 dBm to −90 dBm, corresponding to inputpowers from 10 fW to 1 pW. These 1 dB compression points are comparableto those seen in other SQUID-based microwave amplifiers and 1-2 ordersof magnitude higher than those achieved with typical Josephsonparametric amplifiers. Amplifier dynamic range can be determined bydividing the signal power at 1 dB compression by the noise powercontributed by the SLUG-based amplifier over a given bandwidth.

FIG. 21B is a graph illustrating dynamic range of the SLUG-basedamplifier in accordance with the present invention versus flux forvarious current bias points. FIG. 21B shows two lines corresponding to1.8I₀ (line 205), 1.9I₀ (line 206), and 2I₀ (line 207), with deviceparameters β_(L)=1, β_(C)=0.8, L=10 pH and C=50 fF. In the example shownin FIG. 21B, the zero-temperature quantum spectral density is used forthe shunt resistors of the SLUG device. A typical value of 130 dB Hz isfound, corresponding to a dynamic range of 40 dB in an amplifierbandwidth of 1 GHz. For applications related to dispersive readout ofqubits in a circuit QED architecture, where the focus is on measurementof signals at the level of single microwave quanta in bandwidths oforder 100 MHz to 1 GHz, the dynamic range of the SLUG amplifier,therefore, is more than adequate.

Effect Of Input Circuit Admittance

In the above analysis, the behavior of the isolated SLUG element hasfirst been solved, and the device is treated as a “black box” with knownscattering parameters for the purpose of designing appropriate matchingnetworks. In reality, the nonvanishing admittance at the input andoutput of the SLUG-based amplifier will modify the devicecharacteristics, and a complete treatment must take loading by theexternal circuit into account. The scattering parameters will depend onthe particular realization of the matching network, and a fullexploration of the space of design parameters can become tedious.However, it is found that the performance of the SLUG amplifier is notgreatly affected by the nonvanishing input circuit admittance,particularly once modest steps are taken to decouple the SLUG elementfrom the higher-order modes of the resonant input matching network.

To take into account the admittance of the resonant input matchingnetwork, the junction equations of motion (i.e., equation (10)) can bemodified to include an additional term representing the current drawn bythe input circuit. FIG. 22A is a circuit diagram of a circuit model 210of the SLUG amplifier modified for circuit analysis with finite inputcircuit admittance. The input transmission line 6 having impedance Z₀can be exactly modeled as a pair of coupled, time dependent voltagesources E_(L) and E_(S). These are related to the voltages V_(L,S) andcurrents I_(LS) at the two ends of the transmission line 6 as follows:E _(L)(t)=V _(S)(t−t _(D))+Z _(O) I _(S)(t−t _(D))E _(S)(t)=VL(t−t _(D))+Z ₀ I _(L)(t−t _(D)),   (32)

In equation (32), t_(D) is the propagation delay along the transmissionline 6. The input current is then determined by the additionaldifferential equation:

$\begin{matrix}{{\overset{.}{I}}_{L} = {\frac{1}{L}\left\lbrack {{\frac{\Phi_{0}}{2\pi}\left( {{\overset{.}{\delta}}_{2} - {\overset{.}{\delta}}_{1}} \right)} - E_{L} + {I_{L}Z_{0}}} \right.}} & (33)\end{matrix}$

Using the modified equations of motion for the junction phases, the dccharacteristics of the SLUG amplifier 8 can be calculated. FIGS. 23A are23B are graphs illustrating the I-V and V-Φ curves of a 10 pH, β_(L)=1SLUG amplifier with a 10 GHz quarter-wave input transformer,respectively, with a transmission line input circuit with characteristicimpedance Z₀=2Ω and bare quarter-wave resonance at 10 GHz. In FIG. 23Athree lines are shown corresponding to Φ_(b)=0Φ₀ (line 220),Φ_(b)=0.25Φ₀ (line 221), and Φ_(b)=0.5Φ₀ (line 222). In FIG. 23B threelines are shown corresponding to I_(b)=1.8I₀ (line 223), I_(b)=1.9I₀(line 224), I_(b)=2.0I₀ (line 225). FIGS. 23C and 23D are graphsillustrating the I-V and V-Φ curves of a 10 pH, β_(L)=1 SLUG amplifierwith a 10 GHz quarter-wave input transformer, respectively, with atransmission line input circuit with characteristic impedance Z₀=2Ω andbare quarter-wave resonance at 10 GHz incorporating a 60 pH filterinductor. FIGS. 23C and 23D illustrate that the resonant structure issuppressed through the incorporation of the filter inductor. In FIG. 23Cthree lines are shown corresponding to Φ_(b)=0Φ₀ (line 220),Φ_(b)=0.25Φ₀ (line 221), and Φ_(b)=0.50Φ₀ (line 222). In FIG. 23D threelines are shown corresponding to I_(b)=1.8I₀ (line 223), I_(b)=1.9I₀(line 224), I_(b)=2.0I₀ (line 225).

Referring to FIGS. 23A and 23B, a sharp Shapiro step-like structure isobserved at voltages corresponding to Josephson frequencies that areinteger multiples of the half-wave resonance of the input circuit. Whilequantum fluctuations of the SLUG shunts smooth out this structuresomewhat, it can be desirable to decouple to the SLUG amplifier from thehigher-order standing wave modes of the input circuit, as these modesmay limit amplifier dynamic range and lead to excess noise. Referring toFIG. 22B, to suppress the resonances of the input, it is possible toinsert a filter inductor L_(f) on the order tens of pH between the input6 and the SLUG amplifier 8.

It is now possible to calculate the gain and noise properties of thecomplete circuit of FIG. 22B by performing a full integration of theamplifier equations of motion. Power gain and bandwidth are determinedby driving the amplifier with a sinusoidal input tone and monitoring theoutput of the SLUG amplifier 8 at the excitation frequency. FIG. 24A isa graph illustrating the frequency-dependent gain for the SLUG amplifierwith β_(L)=1, β_(C)=0.8, L=10 pH, C=50 fF, I_(b)=1.8I₀, andΦ_(b)=0.35Φ₀. The input matching network includes a 2Ω transmission linesection with bare quarter-wave resonance at 10 GHz followed by a filterinductor L_(f)=60 pH. FIG. 24A depicts two lines. The first line(element 230) depicts values calculated using the “black box” scatteringparameters of the isolated slug, and the second line (element 232)depicts values calculated by solving the full circuit model of FIGS. 22Aand 22B using a transmission line input with characteristic impedanceZ₀=2 and a length corresponding to a bare quarter-wave resonance at 10GHz, significantly higher than the amplifier operating frequency of 4.5GHz in order to compensate for the additional reactive loading by thefilter inductor.

The agreement with the full circuit simulation is favorable (e.g.,between lines 230 and 232 of FIG. 24A), confirming that the filterinductance has effectively isolated the modes of the SLUG amplifier 8and the transmission line input circuit 6.

To calculate the frequency-dependent noise temperature T_(n)(f), it ispossible to simulate a “hot load/cold load” experiment where the powerspectra S_(V,cold) and S_(V,hot) at the device output are compared forsource resistances at temperatures T=0 and T_(b), respectively. In thethermal regime, this provides equation (34).

$\begin{matrix}{{T_{n}(f)} = {\frac{S_{V,{cold}}(f)}{{S_{V,{hot}}(f)} - {S_{V,{cold}}(f)}}T_{b}}} & (34)\end{matrix}$

In the quantum regime, this provides equation (35).

$\begin{matrix}{\frac{\coth\left\lbrack {{{hf}/2}\;{k_{B}\left( {T_{b} + T_{n}} \right)}} \right\rbrack}{\coth\left( {{{hf}/2}\; k_{B}T_{n}} \right)} = \frac{S_{V,{hot}}}{S_{V,{cold}}}} & (35)\end{matrix}$

The added noise number can then be obtained from equation (26). FIG. 24Bis a graph illustrating the added noise of a 5 GHz SLUG amplifiercalculated with the full circuit model and with the “black box”scattering parameters of the isolated SLUG for the SLUG circuit withβ_(L)=1, β_(C)=0.8, L=10 pH, C=50 fF, I_(b)=1.8I₀, and Φ_(b)=0.35Φ₀. Theinput matching network consists of a 2Ω transmission line section withbare quarter-wave resonance at 10 GHz followed by a filter inductorL_(f)=60 pH. FIG. 24B depicts two lines. The first line (element 234)depicts values calculated using the “black box” scattering parameters ofthe isolated slug, and the second line (element 236) depicts valuescalculated by solving the full circuit model of FIGS. 22A and 22B. Thenoise magnitude is similar in the two cases, although the full circuitsolution predicts a higher frequency for the minimum in the amplifiernoise contribution. The shift in the frequency-dependent noisecharacteristics may be due to a modification of the circulating currentspectral density S_(J) by the nonvanishing admittance of the inputnetwork.

Hot Electron Effects

At millikelvin temperatures electrons decouple from the phonons, and theelectron temperature of the SLUG shunts can be significantly higher thanthe bath temperature. It has been shown that the electron temperatureT_(e) in a metal thin film resistor is given by equation (36).T _(e)=(P/ΣΩ+T _(P) ⁵)^(1/5)   (36)

In equation (36), P is the power deposited in the resistor, Σ is amaterials parameter equal to approximately 2×10⁹ W/m³K⁵, Ω is the normalmetal volume, and T_(P) is the phonon temperature. The elevatedtemperature of the shunt resistors translates directly to elevated noisetemperature of the amplifier. For a device with fixed β_(C), the powerdissipation in the shunts scales as 1/R³. Hot electron effects will beparticularly relevant for the microwave amplifiers discussed here, asoptimal performance is achieved for small SLUG inductance, correspondingto large critical currents and small shunt resistances.

One approach for promoting thermalization of the SLUG shunts atmillikelvin temperatures is to fabricate large-volume normal metalcooling fins in metallic contact with the resistor element. At lowtemperatures, the inelastic diffusion length is on the order of severalmm. Thus, the cooling fins allow hot electrons generated in a localizedregion of the shunt resistor to diffuse over a large volume andthermalize with cold electrons and phonons.

Researchers have demonstrated a significant reduction in the electrontemperature of dc SQUIDs incorporating 400×400 μm² CuAu cooling finswith thickness around 1 μm, with measured electron temperatures under 40mK. A similar approach has been used to suppress hot-electron effectsand reduce the noise temperature of microstrip SQUID amplifiers operatedin the radio-frequency regime. It is possible to integrate normal metalcooling fins with area of order 1 mm² into a standard microwave SLUGamplifier geometry without compromising the microwave integrity of thecircuit. The addition of such cooling fins should make it possible toattain electron temperatures under 100 mK for the device parametersconsidered in this disclosure, corresponding to operation far in thequantum regime for frequencies in the range from 5-10 GHz.

The present disclosure describes a high-frequency amplifier, such as forRF or microwave signals, that may have a SLUG architecture. Someadvantages of the design presented herein over competing approaches tolow-noise, high-frequency amplification of signals havingcharacteristics associated with quantum computing readouts are asfollows:

1. The low-inductance amplifier geometry is compact, straightforward tomodel at RF or microwave frequencies, and readily integrated into atransmission line environment.

2. The amplifier input and output can be well-matched to a 50, 75, orother Ω transmission line impedance, facilitating broadband operation.Moreover, multisection transmission-line input couplers provide a clearpath to attaining bandwidths of order GHz while maintaining excellentgain and noise performance.

3. It is possible to decouple the SLUG modes from the input modes,allowing separate optimization of the gain element and the inputmatching network.

4. The dynamic range of the amplifier is large relative to that requiredfor qubit readout or circuit QED applications.

5. Due to its relatively small magnetic sensing area, the SLUG gainelement is extremely robust and substantially immune to ambient magneticfield fluctuations when compared to other amplifier topologies, such asSQUID-based designs.

The SLUG amplifier presented herein may be attractive in the context ofqubit readout in a circuit QED architecture, either as a nearquantum-limited first-stage amplifier or as an ultralow noisepostamplifier following a Josephson paramp. Other possible applicationsinclude fundamental studies of microwave photon counting statistics, orultralow noise amplification for dark-matter axion detection.

In the present system, multi-section transmission-line input couplersprovide a mechanism for operating at bandwidths on the order ofgigahertz (GHz), while maintaining gain and noise performance thatexceeds that of prior devices. Due to the present system's relativelysmall magnetic sensing area, the SLUG gain element is robust and can beimmune to ambient magnetic field fluctuations. The device's robustnessand immunity to ambient magnetic fields allows for its use in harshenvironments such as in space applications.

In various implementations of the present system, devices having highcritical current density may achieve noise performance approaching thestandard quantum limit in the frequency range from 5-10 GHz, with a gainof about 15 dB for a single-stage device and instantaneous bandwidths onthe order of 1 GHz.

These performance characteristics allow for use of the device in anumber of potential applications including reading-out quantum bits in asuper-conducting or semiconducting quantum information processingnetwork, rapid single flux quantum (RSFQ) superconducting logic/devices,and satellite communications where operating temperatures and requirednoise temperatures are very low.

The present system has been described in terms of one or more preferredembodiments, and it should be appreciated that many equivalents,alternatives, variations, and modifications, aside from those expresslystated, are possible and within the scope of the invention.

The invention claimed is:
 1. A linear amplifier system configured toamplify high-frequency signals, the amplifier comprising: an inputproviding a direct coupling configured to receive a high-frequency inputsignal having a frequency in at least one of a radiofrequency (RF) andmicrowave range; an amplifier including a dielectric material separatingat least two superconducting layers forming an amplifier loop configuredto receive the high-frequency input signal and deliver an amplifiedsignal, the amplifier presenting an inductive short to ground at theinput, and the amplified signal being a linear amplification of thehigh-frequency input signal; and an output providing a direct couplingconfigured to deliver the amplified signal.
 2. The linear amplifiersystem of claim 1 wherein the amplifier forms a superconductinglow-inductance undulatory galvanometer (SLUG) architecture.
 3. Thelinear amplifier system of claim 2 wherein the SLUG architecture isarranged as a symmetric SLUG.
 4. The linear amplifier system of claim 1wherein the amplifier is configured to present a complex input impedancethat is frequency dependent.
 5. The linear amplifier system of claim 1wherein the input includes a tapered transmission line matching sectionconfigured to increase a bandwidth of the amplifier system withoutdegrading a gain of the amplifier system.
 6. The linear amplifier systemof claim 1 wherein the input includes an input transformer havingstepped transmission line impedances selected to increase a bandwidth ofthe amplifier system without degrading a gain of the amplifier system.7. The linear amplifier system of claim 1 wherein the input isconfigured to directly couple to a transmission line impedance.
 8. Thelinear amplifier system of claim 1 wherein the input includes athin-film microstrip connected to a transmission line output.
 9. Thelinear amplifier system of claim 1 wherein at least one of the input andthe output is configured to be coupled to a transmission lineenvironment.
 10. The linear amplifier system of claim 9 wherein thetransmission line environment includes a 50 ohm environment.
 11. Thelinear amplifier system of claim 1 wherein the amplifier is configuredto operate as a phase-insensitive linear amplifier.
 12. A quantuminformation processing network configured to receive and relayhigh-frequency signals, the quantum information processing networkcomprising: a signal source configured to act as at least one of aquantum computing source, a source of qubits, and a linear cavityresonator and at least one of transmitting and reflecting ahigh-frequency signal having a frequency in at least one of aradiofrequency (RF) and microwave range; a transmission line networkconfigured to have an approximately 50 ohm environment and configured toreceive the high-frequency signal; and an amplifier coupled directly tothe transmission line network through an input configured to receive thehigh-frequency signal and an output to deliver an amplifiedhigh-frequency signal, the amplifier being configured to present atleast one of an inductive short to ground at the input, and theamplified high-frequency signal being a linear amplification of thehiqh-frequency signal.
 13. The quantum information processing network ofclaim 12 wherein the amplifier includes a dielectric material separatingat least two superconducting layers forming an amplifier loop configuredto receive the high-frequency signal and deliver the amplifiedhigh-frequency signal.
 14. The quantum information processing network ofclaim 12 wherein the amplifier is arranged in a superconductinglow-inductance undulatory galvanometer (SLUG) architecture.
 15. Thequantum information processing network of claim 14 wherein the SLUGarchitecture forms a symmetric SLUG architecture.
 16. The quantuminformation processing network of claim 12 wherein the transmission linenetwork includes a tapered transmission line matching section coupled tothe input of the amplifier configured to increase a bandwidth of theamplifier without degrading a gain of the amplifier.
 17. The quantuminformation processing network of claim 12 wherein the input includes aninput transformer having stepped transmission line impedances selectedto increase a bandwidth of the amplifier without degrading a gain of theamplifier.
 18. The quantum information processing network of claim 12wherein the amplifier is configured to operate as a phase-insensitivelinear amplifier.